Derivation of the quantum potential from realistic Brownian particle motions

Physics Letters A 162 (1992) 129—136 North-Holland

PHYSICS LETTERS A

Derivation of the quantum potential from realistic Brownian particle motions Piotr Garbaczewski’ Laboratoire de Physique Theorique, Institut Henri Poincaré, UniversitéParis VI, 11 rue Pierre et Marie Curie, F- 75231 Paris Cedex 05, France Received 24 October 1991; accepted for publication 2 December 1991 Communicatedby J.P. Vigier

We present in this Letter a detailed analysis of mechanisms by which the phase space Brownian motion of an ensemble of massive particles, in the diffusion regime, is governed by the Schrodinger equation. It is explicitly shown how the pressure ofthe diffusing ensemble is linked to the quantum potential, known to appear in the Hamilton—Jacobi—Madelung formulation of the corresponding quantum dynamics. The quantum state vector (wave function) corresponds in this picture to the physically real diffusing medium governing the collective evolution of the particle ensemble.

1. Motivation In the framework of Nelson’s reformulation of quantum mechanics [1] the notion of the particle path acquires a well defined meaning on the level of configuration space motions. One can view it as a sample trajectory [2—41followed by the mass point undergoing the Markovian diffusion process in p3 with dynamics constrained by the second Newton law in the conditional mean. While trying to understand this model of quantum phenomena on physically deeper grounds, one is tempted to search for a random phase space propagation, whose configuration space projection would imply stochastic mechanics. This would amount to a phase space derivation of Nelson’s mean acceleration formula, A deep analysis of the links between Einstein— Smoluchowski (configuration space) and Langevin (phase space) descriptions of the Brownian motion was made in the expository reference [5]. As one knows the large friction regime of the Ornstein— Uhlenbeck process allows for the Smoluchowski approximation (e.g. spatial diffusion) in the case of the

general external force, albeit with the applicability limited to the Fick law governed cases. Alternatively, attempts to derive stochastic mechanics in the framework of the so-called stochastic electrodynamics [6,7] indicate that in the Markovian approximation of processes with short correlation times (high friction) one should disregard friction at all to arrive eventually at Nelson’s formalism. It thus seems that the search for a realistic phase space justification of stochastic mechanics ends up with a confusing picture involving both high and low friction limits of Markovian stochastic processes. Our purpose is to view stochastic mechanics (and eventually quantum mechanics) as a special version of the general problem of random flights [8] which arises in the diffusion approximation of the phase space random motion. To get a clearidea ofthe links [91between quantum mechanics and the conventional Brownian motion it seems advantageous to analyse any solvable model in detail. For clarity of discussion (not hampering the generalisations) we shall confine our attention to the case of freely moving Brownian particles in one space dimension.

On leave of absence from: Institute of Theoretical Physics, University of Wroclaw, P1-50205 Wroclaw, Poland. 0375-9601 /92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

129

Volume 162, number 2

PHYSICS LETTERS A

3 February 1992

2. Brownian motion of a single particle

which in the diffusion regime reduces to

The general form of the joint probability distribution W(x, u, t) for a freely moving Brownian particle which at t= 0 begins its motion at x0 = 0 with an

=w(x,

x

arbitrary velocity u0 is derived in ref. [8] under the assumption of the maximally symmetric displacement probability law: 2(FG—H2)]~”2 W(x, u, t)= W(R,S)=[4it GR2—2HRS+FS2 2(FG—H2) (1) xexp(

)‘

—

(8)

1) ~-.

The local (configuration space conditioned) moment of W finally reads x 2t

=

_DVw(.~~t) w(x, t)

(9)

For the second moment we have

Jduu2W(x,u,1)=JdSS2W(R,S)

1)fl~,S_—zu—u

where R=x—u 0(l—e~

0e~’while

2m), F= (2$t_3+4e_/tt_e G=D/3( 1 _e_2Pt), H=D( 1 —e~’)2.

+2uoe_PtJ dSSW(R,S) +u~e2~’dS W(R, S).

J

(2)

/3 is the friction coefficient, D will appear later to be the spatial diffusion constant, D = kBT/ m/3. The marginal distribution of velocities

(10)

In the diffusion regime the leading contribution comes from JdSS2 W(R, S) which can be evaluated by means of handy formulas given in ref. [10]:

J dS

~2

w(R, S) = (FG_H2 +

R 2)

w(u, t)=Jdx W(x, u, t) =(2itG)~”2exp(—S2/2G)=w(S)

x(2~tF)”2exp(—R2/2F)

.

(3)

(11)

In the diffusion regime we thus obtain in the large friction regime (alternatively at times

t

much larger than the relaxation time5’) takes the

<~2>

conventional form m / =(~-~-~) exp~—~ 1/2

w(u, t)

2

(4)

2flt—l) x2\ (D( 21 + ~.1)w(x~ t)

(12)

and the configuration space conditioned (local) moment of W(x, u, 1) takes the form 2> x = (D/3—D/2t) + Ku> (13) ~.

characterising diffusions in velocity space. Analogously the marginal space configuration w(x,


The transport (Fokker—Planck) equation governing

t)=J du W(x, u, t)

the time development of W(x, u, 1) reads ô, W+uV~.W=flV~(Wu)+q/~~W,

=(2i~F)~’2exp(—R2/2F)=w(R)

(5)

in the diffusion regime gives rise to the familiar heat kernel w(x,t)=(47tDt)~’2exp(—x2/4Dt),

(6)

q=D/32

(14)

and implies the local conservation laws o 1w+V(
2>xW)rr —fl~w.

solving Let usô,w=DL~w. now evaluate the first local moment of the joint distribution: =$duuw(x,u,t)

Introducing P(x,t)=(~—~)w(x,t) (16) and assuming that w(x, 1) has no zeros, we can iso-

=w(R) [(H/F)R+u 0e~’]

,

130

(7)

(15)

ôt(xW)+Vx(
late the leading contribution to Pin:

Volume 162, number 2

PHYSICS LETI’ERS A

3 February 1992

3. Brownian evolution of particle ensembles (8, + Ku >~ V)

=

—

/3< u>

—

VP! w.

(17)

In the diffusion regime we have VP —

—

Vw D Vw —flD— + w 2tw

(18)

__,

w

which when inserted into the above momentum conservation law gives rise to the cancellation ofthefriclion term (!): ourprocess effectively appears to befrictionless although in fact operatingin the high friction regime. We have here (8, + ,~V)

D Vw ~

=

(19)

where it is instructive to notice that D Vw !VPosm, P 2WA lnw.

(20)

0smD

All previous derivations refer to a single particle suffering random displacements according to the maximally symmetric probability law, which uniquely characterises the random medium. However [8],itisclearlyallowedinsteadtoimagme a very large number of particles starting under the same initial conditions and undergoing Brownian displacements without any mutual interference according to the same probability law. Such an ensemble cati be built as well of sample flights consecutively executed and having flight duration time 1: even if not perfectly realisable in practice, such an ensemble can be easily produced by means of a computer simulation. Once passing to the ensemble picture, we can consistently follow the lore of the kinetic theory of gases (D/3—D/2t)wasthepressureexerteclatpointx(on and e.g. understand P(x, t)=—(u>~w= the ensemble average!). This concept can be exclu-

Since VPosm/W is irrotational we can easily recover the corresponding potential (the general discussion is given in refs. [1,9])

sively attributed to the swarm ofparticles, where each member independently follows a Brownian motion

Q=Q(x, t)=2D2

ory of diffusion [8] that if w(x, I) denotes the concentration of diffusing substance at x at time 1, then the amount escaping through a given point from the area of larger concentration, per time At is given by —DVw(x, t) At= —U(x, t)w(x, t) At. (23)

w’12

(8, + ,~V)~ =

—

‘

~VPosm =

—

VQ.

(21)

Q(x, t) has the familiar form of the Bohm—Vigier quantum potential (set h/2m = D) except for the posite sign. Irrespective of the sign, however, holds Vw VPosm wVQ. The link of Q and P,~,,,has been lnown for a long time [9,11,12] in connection with the hydrodynamical models of quantum mechanics, where, however, a given particle distribution p(x, t) = I ~u(x, t) 2 is associated with ~

Pq

=

8,p=

—D2pA ln p —

V(pv),

=

—

(8, + vV)v=

—

~

2D2 Ap”2 pl/2

2), D=h/2m. v=2DVln(~/p”

‘

VQq, (22)

according to the same displacement probability law. It is a canonical statement of the macroscopic the-

Here u (x, t) is called [1,5] the osmotic velocity, u=DVw/w. In our case apparently u(x, t) = —x/2t = ~and since Ku> is the local velocity of the particle flow, we realise that (23) does refer indeed to the osmotic transport. In its course the particle swarm expands at the expense of lowering its concentration by the obvious reason: there are more partides leaving the region of higher density than entering it. For a deeper understanding of the previously discussed striking affinity between Q and Qq let us considerthe particle ensemble, whose initial (at time t —

0)

spatial distribution reads p 2) 1/2 exp( —x2/a2) (24) 0(x) = (xa We assume that our particles have approached this distribution in the course of the well defined phase .

131

Volume 162, number 2

PHYSICS LETFERS A

space evolution. Irrespective of its detailed nature (Brownian, classical etc.) we can safely admit that a certain mean velocity field is given in parallel to (25)

KU>~V(X,to)=Vo(X).

We then address the following problem: what is the probability to reach a Ax neighbourhood of a certain point x atfrom timex t> t0, if particles are conditioned to emanate 0 at time t0 with the mean velocity V0The (x0),universal and the propagation Brownian?Brownian maximally issymmetric probability law for spatial displacements AR in the diffusion approximation coincides with the heat kernel expression w(AR)=(4xDAt)~’2 x exp [ (AR )2/4DAt] —

(26)

.

However, what AR is needs to be carefully specified since it depends on the mean velocity value at the reference point (source of particles). Apparently, if we start from x0 with a member of the V0(x0) beam then after time At the particle will reach the point x,

at

3 February 1992

t

0 an arbitrary point x’ with the mean velocity V0(x’) we develop a Brownian evolution anew (this situation has very much in common with the prob1cm of repeated measurements in stochastic mechanics [13]). The corresponding propagation formula (with t0 = 0) reads 2a2Dt)1’2 p(x, t) = dx’ w~(x, t)p0(x’ ) = (4x

J

x

J dx’ exp~ /

—

2

4Dt [x—x’—V0(x’)t]

x’2\ —

—~)~ (30)

Consequently the Brownian evolution of the particle ensemble Po (x) —*p (x, t) quite sensitively depends on the initial mean velocity field, to be contrasted with the single particle Brownian propagation which is insensitive to the initial particle velocity U 0 in the diffusion regime. Hence, how the ensemble propagates due to random fluctuations depends on the phase space preparation procedure which not only

(27)

fixes Po (x) but supplements it with a mean velocity field V0(x). In particular, if the particle ensemble was Brown-

where AR is the purely random contribution. Accordingly we must write

ian prepared Brownian evolution from 2 = 4Dt (through x0 initial velocity U0) then we should set=a0 and arbitrary 0 and the osmotic flow is characterised

AR=x—x0—V0(x0)At,

by V0(x)=x/2t0. Let us consider the ensemble preparation procedure giving rise to p0(x) and also to 2, (31) V0(x)=y2Dx/a

x=x0 + V0 (x,~)At+ AR,

(27’)

thus arriving at the distorted [8,2,3] infinitesimal Brownian propagator. This displacement probability law induces [8] the diffusion equation 8, w(x, t) = V 0 Vw(x, t) + DL~.w(x, t) (28) —

,

which is solved by the Brownian kernel w~0(x,t) = [4xD(t—to) ]_I/2

with yeP’ left unspecified at the moment. The Gaussian propagation integral can be immediately evaluated. We have a

4+4Dta2(l+y)+y24D2t2]}”2

/

xexp~ 4D() ). (29) For simplicity we shall disregard t,, in the above formula, which provides us with the answer to the prob1cm posed before. It is, however, remarkable that our isolated prob1cm can be easily extended (by varying x 0) to the general issue of the Brownian propagation of mitially given particle distributions with non-trivial initial velocity fields. For particles conditioned to pass 132

p(x,

t)=

{x[a

x2a2

~ ex~(— a4+ 4Dta2 (1+ y) + y24D2t2)’ (32) The choice of y= 1 produces the Brownian diffusion of the Brownian prepared V 2 0 (x) = xl 2t0 = 2Dx/a ensemble, while the choice y= 1 apparently leads to (see also ref. [14]) —

Volume 162, number 2

p(x,

t)=

PHYSICS LETTERS A

a [,t(a4+4D2t2)]”2

3 February 1992

Qq refers to the osmotic diffusion process against the initial flow. Remark 1. Eq. (34) is a conventional Schrodinger

xexP(_

(33)

a4±4D2t2)’

which is a well known probability distribution associated with the solution of the Cauchy problem iO, yi(x, t) =

—

DA ~(x,

t),

(34) Indeed ~(x,

J dx’ G(x— x’, (47tiDt) 1/2 J t) =

—

t) çv(x’, 0)

(

~,

(x—x’

exp 2/x)”4(a2+2Wt)”2

=

—

(a x exp( =

—

)2\

4~t

)

0)

~

x2 \ 2 (a2 + 2iDt))

(35)

equation, if we set D=h/2M. Here [4] we can identify M with the mass m of diffusing particles, which fixes the diffusion constant. However, we can as well choose D to be the universal constant and then introduce an effective mass M=h/2D. Remark 2. The Brownian prepared ensemble evolution with y=l Indeed converges theterm y=—l evolution for larger times. thentothe 8Dta2 can be neglected against 4D2t2. Remark 3. If the ensemble preparation procedure is tuned to the background random field to provide p 0(x) with the exact reversal ofthe osmotic flow, then the Brownian evolution of such an ensemble is governed by the equation. This amounts to analysing theSchrodinger diffusion process in terms of average (collective) flows. One flow is directed towards the main concentration, another is born due to the Brownian fluctuations and transports particles away

and I ~v(x, t) 2 =p (x, t) in agreement with (33) The initial (reversed osmotic) velocity field = — 2Dx/a2 is not conspicuously present in (35) but its time evolution is provided by Nelson’s osmotic velocity formula [1]

from the main concentration, hence a diffusion process against the initially given flow appears. The process respects the energy conservation law in the mean:

u(x,t)=DVlnp,

~JdxP(x,t)

u (x,

0) =

—

2Dx a

—~ U (x, t) =

—

2Da2x a4 + 4D2t2’ ________

(36) By (35) we have here developed a particle current with the velocity (called [1,5] current velocity) v(x, t)

= 2DV

v(x,0)=0

in ( ç~i/p”2),

—p

v(x,t)=

4D2tx a4+4D2t2’

(37)

which together with (36) yields ~Jdxp(x,t)(u2+v2)(x,t)=const, compare e.g. ref. [12] where the hydrodynamical discussion is given and ref. [1] for a discussion from the stochastic mechanics viewpoint. The current velocity solves the transport equation 2(Ap”2)/p”2]. (8,+vV)v _VQqV[2D

(38)

(u2+v2)(x,t)=const.

In the course ofthe Brownian propagation two competing flows combine into the mean drift, which asymptotically is dominated by the outgoing current. The quantum potential Qq appears to be a mathematical encoding of such a diffusion process. Remark 4. The initial phase space distribution to which remark 3 applies can be immediately obtained from the Brownian solution (1) if we specialise it to time t 0 and formally replace H by —H. Keeping in the form of W mind that we are2/4D interested in the t0>> fl’ regime and setting t0 = a 0 (x, u) reads 2D/3a2)~2 W0(x,u)=(2it / 4Dx2+8Dxu+2a2u2 X exp~ 4Dfla2 )~ —

(39)

By (3), (5) one easily obtains p 0(x), (26) and ~=DVp0/p0tobe compared with (9). 133

Volume 162, number 2

PHYSICS LETTERS A

4. Brownian evolution of particle ensembles: phase space picture To complete the arguments of section 3 we shall present a parallel phase space description of the Brownian ensemble propagation, for the case 2)x. The more general choice (31) V 0 (x) (2D/a can be=easily investigated by following the pattern. The initial phase space distribution (39) is to be propagated by the Brownian kernel of the form (1), except for another choice ofR and S. Namely we need —

R=x—xo—V

3 February 1992

which implies (46) where U and v are given by (36), (37) respectively. In Nelson’s notation [1] Ku> ~= b(x, t) coincides with the forward drift of the Markovian diffusion process. With the explicit form ofp, 1), U, and b in our hands we can straightforwardly derive the conservation laws respected by our diffusion. The following holds, KU>~=KU>/p(X,t)_—U(X,t)+V(X,t),

0,p=Di~p—V(pb), 0(x0)t—U0(l—e~’)/3’ 0,b+bVb=0

S=u—V0(x0)—u0e~’.

(47)

(40)

and simultaneously We consider dx,,du0 W(x, u,t;x0,u0)W0(x0,u0)

0,p=—V(pv), O1V+VVV_VQq,

~rW(x,u,t).

(41)

In the diffusion regime the u0 contributions can be neglected, and the effective du0 integration yields W(x, u, t) = dxi, W(x, u, t; X0 )Po (x0) . (42)

J

Apparently (compare e.g. (29), (30))

$

dU W(x,

U,

t)=j’ dU $ dxc, W(x, u, t; x0)p0(x0)

=J dx0 w~0(x,t)p0(x0)=p(x,

t).

(43)

We can the local expectation value with respect to evaluate u by the same interchange ofthe integration order procedure: Ku>

=$ du uW(x, U, t)

=J dx~J du uW(x,

U, t;

x

0)p0(x0) =J~oVo(xo)w~o(x,t)po(xo). Setting

(44)

—

(2D/a

0 we can proceed fur-

ther to arrive at 2D —

—i

(4x2a2Dt)I/2$

KU>~=x/2r=—U(x,t),

(50)

where apparently 0~w=—V(Ku>,w)

=—VQ(x,r)=VQ0(x,t) and

—

—

space (Brownian) The striking between Q and Qq arguments. governed diffusions (cf. affinity section 2) becomes here transparent, if we make a time substitution 2 + 4D2t 2/a2, (49) 4Dr= a which amounts to changing the clock (t—~~(t)) according to which the stochastic process is executed. Eq. (49) implies (the notations of section 2 are strictly followed) 2 exp( —x2/4D~), p(x, t)= w(x, r)= (4xDt)”

(o~+Ku>~v)Ku>~

dx0x0

2 2)x 2 [x—x0+(2D/a 0t] x0\ —~i) xexp( 2—2Dt)4Dt 2Dx(a a4+4D2t2 p(x, t), — —

which completely characterises the diffusion process. We would like to emphasise the classical (Liouville) propagation formula for b(x, t) = KU> in the absence of external forces. We have thus arrived at the notion of the quantum potential (Qq) governed diffusion process, by means of purely phase

2 )x

V 0 (x0) =

KU>=

(48)

(45)

Q(x, x) =2D2

(51)

_____

1/2

(x, x)

=

—Q 0(x,

134

t),

Volume 162, number 2

PHYSICS LETTERS A

‘~‘osm(x,

r) =(D2wA ln w)(x, r)=—P

0(x, t). (52) 2 we relate (50) with the initial By setting 4Dr0= distribution (24).aThe following holds, w(x,

t)=J

~‘

3 February 1992

is inconsistent with this picture. Quantum particles really move in a “hidden thermostat”, to recall de Broglie’s famous suggestion. [19]. On the other hand we give further support to the idea that quantum mechanical particles are transported along realistic space time trajectories with re-

w(x—x’, t—r 0)w(x’,

T~)

(53)

which is a propagation formula characteristic for free diffusion. Hence, while Qq (x, t) governs the Brownian diffusion against-the-flow, within a specifically prepared (conditioning!) particle ensemble, it appears that Q(x, r) = — Qq(x, t) governs a conventional, unrestricted (no conditioning) diffusion process of section 2, albeit with respect to another clock, (49). The phase space implementation is obvious. Diffusion processes (48) and (51) are here inseparably connected and live simultaneously, providing a suggestive illustration of the action—reaction principle. The “quantum pressure” Pq (1) is exactly balanced by the osmotic pressure Posm (t( t)), the osmotic flow Ku>~(x, r) balances the remainder u(x, t) of the initial particle the 2+ v2 flow. )p(x, Together 1) = const with it seems conservation law fdx (u to establish a link with the ideas of ref. [15], where particle—medium interaction was acting both ways (action—reaction) with a strictly observed energy— momentum conservation law at each minute scat-

alistic momenta in plain opposition to the Copenhagen convention. Quantum state vectors (wave functions) correspond in this model to real physical phenomena which are governed by the background random field. They encode the mean (collective) data of the diffusion process. By (26)— (29) one can here introduce a concept of the stochastic control fields. Indeed, in the description ofparticle ensembles we have finally arrived at the field of locally defined displacement probability laws, which control the diffusion in the infinitesimal neighborhood of each given point. One can thus tell that ~v(x, t) is related to the specific control field (state of the diffusing medium) from which the quantum mechanical statistics (e.g. Born postulate) does originate. Acknowledgement I have benefited from discussions with Professor J.P. Vigier and his hospitality extended to me in Paris.

tering event.

References 5. Conclusions In the above context of the essentially Brownian implementation of the quantum mechanical time development, let us stress the fundamental importance ofboth theoretical and experimental investigation of the particle trajectory concept in the realm of quantum theory [16—18].In our discussion particle trajectories (“hidden variables”) primarily arise as phase space Brownian paths, hence they refer to the non-trivial energy—momentum exchanges between the particle and the surrounding diffusing medium;

the quantum potential refers thus in the diffusion regime to a quite realistic physical phenomenon of

random accelerations. Consequently any idea of “ghost information” or “empty wave” propagation

[1] E. Nelson, Quantum fluctuations (Princeton Univ. Press, Princeton, 1985). [2] N. Cufaro Petroni, Phys. Lett. A 141 (1989) 370. [3] P. Garbaczewski, Phys. Lett. A 143 (1990) 85. [4] P. Garbaczewski, Phys. Lett. A 147 (1990)168. [5] E. Nelson, Dynamical theories of the Brownian motion

(Princeton Univ. Press, Princeton, 1967). [6]L. de la Peila and A.M. Cetto, J. Math. Phys. 18 (1977) 1612. [7] P. Claverie and S. Diner, Int. J. Quantum Chem. 12, Suppl. 1 (1977) 41. [8] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. [9] P. Garbaczewski, Nelson’s stochastic mechanics as the problem of random flights and rotations, in: Karpacz 91 Proc. on Nonlinear fields, classical, random, semiclassical,

eds. P. Garbaczewski and Z. Popowicz (World Scientific, Sin~pore,1991). [10] J. Rzewuski, Field theory, Vol. II (Iliffe, London, 1969). 135

Volume 162, number 2

PHYSICS LETTERS A

3 February 1992

[11 ]T. Takabayasi, Progr. Theor. Phys. 11(1954) 341.

[17] H. Rafli-Tabar, Phys. Lett. A 138 (1989) 353.

[l2]E.H.Wilhelm,Phys.Rev.D 1(1970)2278. [13] Ph. Blanchard, S. Golin and M. Serva, Phys. Rev. D 34 (1986) 3732. [14] K. Namsrai, Nonlocal quantum field theory and stochastic quantum mechanics (Reidel, Dordrecht, 1986). [15] D. Bohm and J.P. Vigier, Phys. Rev. 96 (1954) 208. [16] N. Cufaro Petroni and J.P. Vigier, Single particle trajectories and interferences in quantum mechanics, Found. Phys., to appear.

[l8]J.P. Vigier, in: Proc. 3rd tnt. Symp. on Foundation of quantum mechanics, Tokyo, 1989, pp. 140—152. [19] L. de Broglie, La thermodynamique de Ia particule isolée (ou thermodynamique cachée des particules) (GauthierVillars, Paris, 1964).

136